The Merrifield-Simmons Index And Hosoya Index Of Some Graphs

Merrifield-Simmons index and Hosoya index are the two valuable topological indices in chemical graph theory. The Merrifield-Simmons index σ(G) of a graph G is defined as the total number of the independent sets of the graph G and the Hosoya index μ(G) of a graph G is defined as the total number of the matchings of the graph G. These two topological indices have important applications in making new compounds and new drugs.In this paper, firstly, based on the definition of polyphenyl chains、six-membered ring spiro chains、hexagonal chains、phenylene chains and so on, the elements for the development,6-order cycle expanded to n-order cycle, consists of several the same order cycles sequence by "one vertex coincide"、"one edge connect"、"one edge coincide"、"adjacent two vertices corresponding connect" four special connect ways, we construct four classes special graphs G;(m,k), i=1,2,3,4, respectively, study their number with respect to the Merrifield-Simmons index and Hosoya index in the non-isomorphic connected positions, and their formulas are given.Secondly, according to the definition of corona of two graphs, based on the existed results for corona graphs PnοK1、PnοK2、CnοK1、Cn ο K2、Pn ο Kt、 Cnο K1with respect to Merrifield-Simmons index and Hosoya index, further study the number of two classes special corona graphs Pnο and Cn οH with respect to the Merrifield-Simmons index and Hosoya index, and get some more general results.Finally, by three cycles through the above four special connected ways, construct four kinds of special graphs Ti(k), i=1,2,3,4, respectively, and characterize the ordering of these special graphs Ti (k) with respect to Merrifield-Simmons index and Hosoya index. The results show that the graphs Ti(k) about the two topological indices on the contrary, for each i∈{1,2,3,4}. the specific results are as following. σ(Ti(1))<σ(T,(3))<…<σ(Ti(2s+1))<σ(Ti(2s+2t))<…<σ(Ti(4))<σ(Ti(2)). μ(Ti(1))>μ(T,(3))>…>μ(Ti(2s+1))>μ(Ti(2s+2t))>…>μ(Ti(4))>μ(Ti(2)). where s≥0,...